Stochastic homogenization of subdifferential inclusions via scale integration

Abstract

We study the stochastic homogenization of the system

−   ση ∈ ∂φη η ∈ ∂φη η ∈ ∂φηη ∈ ∂φηdiv ση = fη(∇uη),

where

φη is a sequence of convex stationary random fields, with p-growth. We prove that sequences of solutions (ση, uη) converge to the solutions of a deterministic system having the same subdifferential structure. The proof relies on Birkhoff’s ergodic theorem, on the maximal monotonicity of the subdifferential of a convex function, and on a new idea of scale integration, recently introduced by A. Visintin.